# Monday April 13th ## Non-Archimedean Uniformization of Abelian Varieties We'll start by recalling the Archimedean uniformization theory, i.e. over $\CC$: every compact connected $\CC\dash$analytic (thus commutative) Lie group is isomorphic to $\CC^g / \Lambda$ a discrete cocompact lattice isomorphic as a $\ZZ\dash$module to $\ZZ^{2g}$. Note that $\Lambda$ is a complex torus, and not a linear torus $\GG_m^n$. > Thus all complex torii are isomorphic as topological (even real Lie) groups. Note that if $(K, \abs{\wait})$ is a complete non-Archimedean field, then full lattices in $k$ do not exist in characteristic zero, so we want a multiplicative version. The right notion of isomorphism to use here is homothety (scaling by a complex unit) and thus $\Lambda \subset \CC^g$ has an action of $\GL_n(\CC)$ so $\CC^g/\Lambda \cong \CC^g/m \Lambda$, so we may choose $\Lambda$ to have a $\ZZ\dash$basis given by $\lambda_1 = e_1, \lambda_g = c_g$ generalizing the usual basis of $1, \tau$ in the 1-dimensional case. We have an exponential map $\operatorname{exp}: \CC^g \to (\CC\units)^g$ where $\vector z \mapsto e^{2\pi i \vector z}$, so pushing $\Lambda$ through $\operatorname{exp}$ yields a full lattice $\Lambda' \subset (\CC\units)^g$ where $\CC^g / \Lambda \cong (\CC\units)^g/\Lambda'$. In $\dim g > 1$, not every $\CC\dash$torus corresponds to an abelian variety (contrasting the $g=1$ case). Changing basis for $\Lambda$ yields a matrix $(I \mid Z)$ where $Z \in \Mat_{g, g}(\CC)$, and Riemann showed that $\CC^g / \Lambda$ determines an abelian variety iff this matrix is symmetric with positive definite imaginary part. This occurs iff $\Lambda$ admits a *Riemann form*, which is an analog of the polarization form, and is given by $H: \CC^g \cross \CC^g \to \CC$ that is Hermitian and positive definite such that $H(\Lambda) \subset \ZZ$. > Equivalently, having an ample line bundle, some multiple of which embeds it into some projective space. By counting parameters, we can determine the dimension of the moduli space of $g\dash$dimensional complex torii: since we have a $g\times g$ nonzero matrix, this yields $g^2$. Similarly, counting moduli for abelian varieties yields $g(g+1)/2$. Note that these coincide for $g=1$ but differ for $g>1$, so there's an entire dimension of torii that aren't abelian varieties. We define a $K\dash$analytic torus as $T(\Lambda) = (K\units)^g / \Lambda$. This is a $g\dash$dimensional $K\dash$analytic manifold which is compact iff $K$ is locally compact. Question: For which $\Lambda$ is there an abelian variety $A/K$ such that $A \cong T(A)$? This should mean that for all $K \subset L \subset \bar K$, we have $A(L) \cong (L\units)^g / \Lambda$. But which category should we take this in? We need to ensure that the notion of isomorphism isn't so weak that these isomorphism occur for trivial reasons. We'll work in the category of rigid $K\dash$analytic spaces: locally ringed spaces (with a sheaf of functions) equipped with a Grothendieck topology, which are locally isomorphic to *affinoid* spaces. The most basic example of an affinoid space: $\ZZ_p \injects \QQ_p$, locally compact but not compact, taking the $p\dash$adic disc (defined by an inequality in the norm) is affinoid. In general, points in a space satisfying norm inequalities and thus bounded, generalizing what happens in $p\dash$adic fields. > Affinoid spaces will be analogous to CW complexes. A Grothendieck topology (like the etale topology) will be a generalization of open sets to arbitrary categories by saying what it means for an object to cover another. These restrict the coverings allowed by the general topology to those that appear in algebraic geometry. Remark: A connected rigid $K\dash$analytic space $X$ has a in fact has a *field* of meromorphic functions $K(X)$, which is an important invariant. Example: Take a connected complex manifold, we can consider its meromorphic functions which is a sheaf that is locally given by quotients of holomorphic functions. For $\Lambda \subset (K\units)^g$, the transcendence degree of $K(T(\Lambda))$ over $K$ is finite and $\leq g$ because $T(\Lambda)$ is a proper (similar to compact) $K\dash$analytic spaces. There is a correspondence between curves and function fields, which we get whenever the transcendence degree is 1. So this yields a good analog for complex torii. Recall that if $T(\Lambda)$ is a complex torus implies that $0 \leq \deg_\CC^{\text{transc.}} \CC(T(\Lambda)) \leq g$, then there exists a universal quotient $A_n$ with $T\surjects A_n$. Let $\Lambda = (K\units)^g$ and $\lambda\dual = \hom_\ZZ(\Lambda, \ZZ)$, then $T(\Lambda)$ is an abelian variety iff there exists a map $\sigma: \Lambda \to \Lambda\dual$ such that $\sigma(\gamma_1)(\gamma_2) = \sigma(\gamma_2)(\gamma_1)$ and $(\gamma_1, \gamma_2) \definedas - \log \abs{ \sigma(\gamma_2)(\gamma_1)}$ is positive definite. Assume no that $K$ is a discretely valued valuation field, and abbreviate the identity component of the Néron special fiber as the Néron group ($g\dash$dimensional algebraic group over the special fiber which is an abelian variety iff good reduction) Theorem : \hfill 1. If $T(\Lambda)$ is an abelian variety, this it has split multiplicative reduction and the Neron group is isomorphic to $\GG_m^g / \Lambda$ 2. Conversely, if $A/K$ has split multiplicative reduction, then $A$ is isomorphic to an analytic torus. We define $S/K$ to be a *semi-abelian* variety if $0 \to T \to S \to B \to 0$ where $T$ is a linear torus of dimension $u(s)$ and $B$ is an abelian variety of dimension $\beta(s)$. Theorem : If $(K, \abs{\wait})$ is a complete non-Archimedean field and $A/K$ a $g\dash$dimensional abelian variety then there exists a semi-abelian variety $0\to T\to S\to B \to 0$ of dimension $g$ which has potentially good reduction and a $g_K\dash$module $M \cong_\ZZ \ZZ^{u(s)}$ and $0 \to M \to S(K\sep) \to A(K\sep) \to 0$ where $\rk_\ZZ M^{g_k}$ (the invariants) is the split toruc ? of the Neron group of $A$ and for all $L/K$ the Néron groups of $S/L$ and $A/K$ are isomorphic. > Note that we're taking the Néron group of a semi-abelian variety here, which is a generalization.